Inverse distributions: the logarithmic case

Dario Sacchetti

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 4, page 785-795
  • ISSN: 0010-2628

Abstract

top
In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.

How to cite

top

Sacchetti, Dario. "Inverse distributions: the logarithmic case." Commentationes Mathematicae Universitatis Carolinae 39.4 (1998): 785-795. <http://eudml.org/doc/248279>.

@article{Sacchetti1998,
abstract = {In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.},
author = {Sacchetti, Dario},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {natural exponential family; Laplace transform; variance function; logarithmic distribution; inverse distribution; variance function},
language = {eng},
number = {4},
pages = {785-795},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Inverse distributions: the logarithmic case},
url = {http://eudml.org/doc/248279},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Sacchetti, Dario
TI - Inverse distributions: the logarithmic case
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 4
SP - 785
EP - 795
AB - In this paper it is proved that the distribution of the logarithmic series is not invertible while it is found to be invertible if corrected by a suitable affinity. The inverse distribution of the corrected logarithmic series is then derived. Moreover the asymptotic behaviour of the variance function of the logarithmic distribution is determined. It is also proved that the variance function of the inverse distribution of the corrected logarithmic distribution has a cubic asymptotic behaviour.
LA - eng
KW - natural exponential family; Laplace transform; variance function; logarithmic distribution; inverse distribution; variance function
UR - http://eudml.org/doc/248279
ER -

References

top
  1. Barndorff-Nielsen O., Information and exponential families in statistical inference, Wiley, New York, 1978. MR0489333
  2. Dieudonné, J., Infinitesimal Calculus, Houghton Mifflin, Boston, 1971. MR0349286
  3. Fichtenholz G.M., Functional Series, Gordon and Breach, Science Publishers, 1970. Zbl0213.35001MR0463743
  4. Guest G., Laplace Transform and an Introduction to Distributions, Ellis Horwood, 1991. MR1287158
  5. Johnson N.L., Kotz S., Discrete Distributions, Houghton Mifflin, Boston, 1969. Zbl1092.62010MR0268996
  6. Jorgensen B., Martinez J.R., Tsao M., Asymptotic behaviour of the variance function, Scand. J. Statist. 21 (1994), 223-243. (1994) MR1292637
  7. Letac G., La reciprocité des familles exponentielles naturelles sur , C.R. Acad. Sci. Paris 303 Ser. I2 (1986), 61-64. (1986) MR0851270
  8. Letac G., Lectures on natural exponential families and their variance functions, I.M.P.A., Rio de Janeiro, 1991. Zbl0983.62501MR1182991
  9. Letac G., Mora M., Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990), 1-37. (1990) Zbl0714.62010MR1041384
  10. Mora M., Classification de fonctions variance cubiques des familles exponentielles sur , C.R. Acad. Sci. Paris Sér I Math. 302 (1986), 587-590. (1986) MR0844163
  11. Morris C.N., Natural exponential families with quadratic variance functions, Ann. Statist. 10 (1982), 65-80. (1982) Zbl0498.62015MR0642719
  12. Sacchetti D., Inverse distribution: an example of non existence, Atti dell' Accademia delle Scienze Lettere ed Arti di Palermo, 1993. 
  13. Seshadri V., The Inverse Gaussian distribution, Oxford University Press, Oxford, 1993. Zbl0942.62011MR1306281
  14. Tweedie M.C.K., Inverse statistical variates, Nature 155 (1945), 453. (1945) Zbl0063.07892MR0011907

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.